Experimental Evidence of a Low Energy Seebeck Effect Anomaly

Today, I will discuss a peculiar experimental result which has been found by a group at CalTech. They are measuring the Seebeck coefficient for materials at high temperatures. The Seebeck effect describes the equilibrium of voltage and thermal gradients in the case where there is no current, or a steady state is eventually reached. The Seebeck effect states:

\nabla V = -S \nabla T,

where V is the electric potential, T is the temperature, and S is the Seebeck coefficient. S is dependent on the properties of the material and T. Metals have modest Seebeck coefficients and some semiconductors have larger coefficients. Newly discovered materials with higher coefficients would improve energy extraction efficiency from natural temperature gradients, such as the engine of a car or a refrigeration system. Fully understanding the Seebeck effect in the low gradient limit is essential for optimization of such designs.

The CalTech paper (http://scitation.aip.org/content/aip/journal/rsi/82/6/10.1063/1.3601358) finds a rather bizarre non-linearity in accurate measurements of the Seebeck effect. The non-linearity is simply an affine transformation, or shift by a constant \alpha. This constant is also dependent on the material and the temperature, which would change the Seebeck effect to this:

\nabla V = -S\nabla T + \alpha

This seems rather unintuitive! Why would a voltage spontaneously be created if both sides were at the same temperature? Also, if one put a heat reservoir of the same temperature on each ends of a metal bar, it would create a small voltage of about 20 microvolts. This is not some random fluctuations or oscillations from side to side like Johnson or Shot noise, this is truly a DC measurement. Clearly, to describe this effect accurately, more physics is needed in understanding what determines this unknown constant \alpha .

What other possible occurrences in nature could cause this force? Perhaps gravity can play a role. It is weak, but this experiment is probing the smallest electromagnetic measurements currently possible. This is essentially a low energy, or IR “divergence”. Let’s think about how gravity could possibly effect temperature.

The first thing that may come to mind is the Unruh effect, which states than an accelerating object would observe black body radiation from the vacuum. Therefore, by the equivalence principle, the force of gravity on earth causes an accelerated frame with respect to the Unruh effect. So the closer one gets to the surface of the earth, the more radiation one would see, In other words, everything measured around it would seem to be hotter than it actually is. To disagree with a temperature gradient caused by gravity would disagree with fundamental principles of relativity.

However, this Unruh effect is so small that it would be about 10^{-40} times smaller than a comparable Seebeck effect. It would be undetectable by modern voltage measurement resolution. So maybe gravity is not the answer for the mysterious \alpha. But wait, the Seebeck effect is a classical effect, so shouldn’t there be a classical thermogravity effect just as there is a classical thermoelectric effect?

This effect does not seem to be accepted by the majority of physicists. Fairly recent papers by Coombes and Laue in 1984 as well as Roman in 1996 argue why this thermogravity effect should not exists, as a microcanonical ensemble suggests that the stationary state is always at the same temperature throughout. After all, this is the definition of equilibrium. While this is true for the simplest of systems, I am skeptical if it can be proven that it must occur for any general system.

Measurements of the atmosphere show that there exists a temperature gradient which could be due to gravity. While the system is not stationary and highly non-adiabatic, meteorologists and weather analysts approximate the atmosphere to be an adiabatic system and use a classical analogue of the thermoelectric effect, except essentially using mass instead of electric charge in the coefficients. Experiments done by Roderich Graeff has measured this effect and uses the classical adiabatic equation to match fairly accurately with his findings.

However, the CalTech experiment shows that a voltage gradient is expected to be measured with thermocouples (which exploit the Seebeck effect) when there is no temperature gradient. Perhaps there is no temperature gradient and we simply have not understood this anomalous \alpha. Who is to say if we have measured a gravitational effect, or are noticing this bizarre non-linearity?

A simple experiment which I have thought of will tell us exactly how much of \alpha is due to gravity and how much is due to unknown physics. Imagine a metal bar which is held to have zero temperature gradient on both ends, just as was done at CalTech. If \alpha is measured to be isotropic, then it cannot be due to gravity. If \alpha (the voltage measurement) is zero when the bar is horizontal and a maximum when vertical, then it would suggest that gravity is a valid explanation. Furthermore, if the bar is rotated by 180 degrees, one would expect the voltage gradient to reverse signs if this was a gravitational effect. The final possibility is that gravity is a relevant factor, but not telling the whole story. This would occur if \alpha is some minimum non-zero absolute value when in the horizontal position and at a maximum absolute value when in the vertical position.

I plan to ask the researchers at CalTech to see if they have tried rotating their experiment by 90 degrees and noticed a difference. If \alpha is isotropic, then Graeff has not measured a gravitational effect, but rather an anomaly of the Seebeck equation in the low differential energy limit. If gravity does play a role, a quantum statistical derivation of the effect is the next logical step. I will start this pursuit by assigning the following.

Ideally, we would like to renormalize our equation in such a way to force it to be linear. Fundamentally, the true Seebeck effect should be linear. Instead of working with \alpha, it will be convenient to define an arbitrary physical field \Gamma, such that \nabla\Gamma = \alpha. Note that in general, \gamma is dependent on the material properties and the temperature. Next, a Legendre transform on the potential V is made: U = V - \Gamma. Now, we have recovered a truly linear Seebeck effect which is dominated by electromagnetism for ordinary laboratory voltages:

\nabla(V - \Gamma) = \nabla U = -S\nabla T

But what is \Gamma physically? If it is purely due to gravity, it should be something like \Gamma = \epsilon \Phi, where \Phi is the gravitational potential and \epsilon is some material and temperature dependent coupling constant relating the strength between the electrogravity and thermogravity effect. While \epsilon does have units, it should have a value much less than 1 for the magnitudes typical lab experiments.

\epsilon has basically been written down by Loschmidt in the 1860’s when he argued that gravity would cause a perpetual mobile of the second kind. Maxwell and Boltzmann shut Loschmidt down by saying this process was violating the second law. It is my personal opinion that both sides of this argument were not exactly correct. It seems that there is new fundamental physics lying beyond us. Perhaps this discrepancy will lead to new types of efficient engines which could be created at a microscopic level. Once this process is understood, technology could be advanced to magnify such an effect.

Loschmidt’s argument led to the classical equation which meteorologists use incorrectly for describing the atmosphere, which is non-adiabatic. The thermoelectric effect has a more rigorous quantum statistical mechanics derivation of the Seebeck coefficient. I am currently making some progress on pursuing an analogous quantum statistical derivation of \epsilon, which would provide a theory for a class of materials to be experimentally verified.

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